Does (prostatic fluid/precum/madhiy etc.) require washing the part of the cloth stained? Does it require washing the private part?

An overview of the scholarly views

First of all we need to clarify the scholarly view on madhy: Is it tahir (pure) or is it najis (impure)?

• The majority view (some scholars even said it is a consensus) is that it is najis especially based on the ahadith you've quoted saying that one should sprinkle a handful water on what is smeared with it and wash the private parts (not only sprinkle water on them). These both statements make clear that it can't be tahir.
  Note that some scholars among them imam al-'Ayni بدر الدين العيني the commentator on Sahih al-Bukhari pointed at the fact that Arabs use:

النضح and الرش

referring to washing not simply sprinkling! This requirement is also clearly quoted in some ahadith like in here al-Muwatta'.
Nevertheless the main view is that madhy or the urine of a baby (boy) who has not yet been fed with "normal" food are actually najis, but their najasah is rather light compared to urine in general etc.

• However one rather deviate view of imam Ahmad exists which says it is tahir (note that the applied and documented view in his madhhab is the opposite view).
  Note: If I found time and after a research I will address the ahadith supporting the deviate view (I just have a feeling they could be explained as special cases).

Moreover some scholars explained the reason why one should wash the private parts saying it stops it from coming out like al-Aabadi was cited in his 'Awn al-Ma'bud عون المعبود (commentary on Sunan abi Dawod) referring when to this hadith:

^{In the following and elsewhere in this post -if not claimed otherwise- I'll translate from Arabic sources, as these translations are of my own: take them with the necessary care!}

وَيُقَال إِنَّ الْمَاء الْبَارِد إِذَا أَصَابَ الْأُنْثَيَيْنِ رَدَّ الْمَذْي فَلِذَلِكَ أَمَرَهُ بِغَسْلِهَا.
It is said that cold water if it came in contact with his testicles would stop pre seminal fluid that's why he was ordered to wash them. (Source: 'Awn al-Ma'bud on islamweb)

Some scholars etc. also addressed the fact that madhy and maniy (semen) have similarities (in their properties, causes and attributes and further madhy is considered as part of maniy), but these similarities could hardly be a reason for the same ruling, as we know that semen emission requires ghusl!

Additional info

In Arabic the word for pre seminal fluid for male is called madhy

المذي

While for the female it is called qadhy

القدي

For further information about the above you may refer to the fatwa islamweb.net #169521 (in Arabic) which discusses in details the matter and tries to emphasize on the superiority of the majority view.

Is it alright to take the easier view?

The matter is not that easy: As the imams ibn Seereen and Malik have said:

"This knowledge is religion, so be careful from whom you take your religion." (See for example Did Imam Malik describe the hadith sciences as "this science is a religion"?)

One shouldn't blindly follow, but follow what one understands and accepts for himself especially if one finds the evidences and explanation helpful and find peace of mind (and heart) if following it.

On the other hand we may find a statement of Sufyan a-Thawry saying:

(Real) Knowledge with us (in our opinion) is a permission from a trustworthy person, while rigour can be made (declared) by anybody (See for example How to interpret Sufyan a-Thawri's statement on permission in fatwas?)

In a fatwa asking the same question as you asked above they quoted from Fatawa (or Fatawi) 'Ilysh عليش (the maliki scholar) saying -more generally-:

إن خروج المقلد من العمل بالمشهور إلى العمل بالشاذ الذي فيه رخصة من غير تتبع للرخص صحيح عند كل من قال بعدم لزوم تقليد الأرجح، وهو قول الأكثر من الأصوليين، ويباح للمقلد أن يقلد من شاء من أقوال المجتهدين، وإن نقل الإجماع على منع ذلك غير صحيح
If a muqallid (a follower of a madhhab) leaves applying the common (view) to applying the deviate view which contains an ease (relaxation) without (generally) following relaxations is considered as correct by all those who consider not practicing taqleed the superior view. And this is the view of most of scholars of Osol (al-Fiqh), and it is permissible for the muqallid to follow whom ever he wants among the Mujtahids and if anybody pretended a consensus of the opposite of this it is incorrect.
(Source: fatwa islamweb #195377)

After saying (Quote from the fatwa #195377 above):

وأما هل يجوز لك أن تأخذ بالقول القائل بطهارة المذي إن تبين لك صحته، فالجواب أن نعم، إن تبين لك صحته، ولم يكن من باب التشهي وتتبع الرخص.
As to your question: whether it is permissible to take (follow) the view of those saying that madhy is pure, if you are convinced by its correctness. The answer is yes, if you are convinced and you didn't follow it duo to affection or the goal to follow relaxations.